280 XIV INTEGRATION IN LOCALLY COMPACT GROUPS (14.8.1.1) The image of the measure /x®£ under the homeomorphism (s, x)\-+(s, s~*x) ofG2 onto G2 is the measure \JL ® ft. We may assume that n ^ 0. Then, for every F e j?T+(G2), the function (s, x)\-+F(s, s'tx) belongs to jf+(G2), and we have F(s, s~1x) dn(s) dft(x) = dii(s) F(s9 s~*x) dft(x) JJ J J (s9x)dff(x) by virtue of the left-invariance of ft and the Lebesgue-Fubini theorem. This proves the lemma. Now suppose that p and /•/? are convolvable. Since |/*/?| = |/| •/? (13.13.4) we may limit ourselves to the case where p. ^ 0 and/^ 0. For each function he Jfc(G), the function (s9x)t-+h(sx)f(x) is (/x(g)j5)-integrable by hypothesis (13.21.16 and 3.14.3); by virtue of (14.8.1.1), the same is true of the function (jf9x)^h(x)f(s'1x) (13.7.10). If Afc is the set of points jceG such that h(x) 7* 0, it follows from the theorem of Lebesgue-Fubini that there exists a jS-negligible set NA in A/, such that, for each xeAHr\ ([Nft, the function st-*f(s~1x) is /z-integrable. Taking a sequence of functions h in JT+(G) such that the corresponding sets Ah cover G (4.5.2), we see that the function st-*f(s~1x) is /^-integrable except at the points x of a ^-negligible set N. Furthermore, it follows from the Lebesgue-Fubini theorem that the function x\r-+h(x) \f(s~1x) dp,(s)9 defined almost everywhere with respect to /?, is )S-integrable, and that dp(x) = (h(x) dfi(x) tf(s-lx) This proves that the condition of (14.8.1) is necessary, and that n * (/• ft) = £•£(13.13.1). Conversely, suppose that the condition is satisfied. We have to show that, for each function /ze^+(G), the function (s,x)*~>h(sx)f(x) is(/*®/?)- integrable. Now this function is (/x ® /?)-measurable because h is continuous (13.21.13), and therefore it is enough to show that JJsufficient that there should exist a ^-negligible set N such that the